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toral lie algebra | In mathematics, a toral subalgebra is a Lie subalgebra of a general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field). Equivalently, a Lie algebra is toral if it contains no nonzero nilpotent elements. Over an algebraically closed field, every toral Lie algeb... | wikipedia |
toroid | In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow rectangle-section ring is produced. If the revolved fig... | wikipedia |
toroid | The term toroid is also used to describe a toroidal polyhedron. In this context a toroid need not be circular and may have any number of holes. A g-holed toroid can be seen as approximating the surface of a torus having a topological genus, g, of 1 or greater. | wikipedia |
toroid | The Euler characteristic χ of a g holed toroid is 2(1-g).The torus is an example of a toroid, which is the surface of a doughnut. Doughnuts are an example of a solid torus created by rotating a disk, and should not be confused with toroids. Toroidal structures occur in both natural and synthetic materials. | wikipedia |
transcendental functions | In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically using a finite amount of terms. Examples of transcendental func... | wikipedia |
mathematical transformations | In mathematics, a transformation is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transfor... | wikipedia |
generating function transformation | In mathematics, a transformation of a sequence's generating function provides a method of converting the generating function for one sequence into a generating function enumerating another. These transformations typically involve integral formulas applied to a sequence generating function (see integral transformations)... | wikipedia |
generating function transformation | + f 1 1 ! z + f 2 2 ! z 2 + ⋯ . | wikipedia |
generating function transformation | {\displaystyle {\widehat {F}}(z)=\sum _{n=0}^{\infty }{\frac {f_{n}}{n! }}z^{n}={\frac {f_{0}}{0! }}+{\frac {f_{1}}{1! | wikipedia |
generating function transformation | }}z+{\frac {f_{2}}{2! }}z^{2}+\cdots .} In this article, we use the convention that the ordinary (exponential) generating function for a sequence { f n } {\displaystyle \{f_{n}\}} is denoted by the uppercase function F ( z ) {\displaystyle F(z)} / F ^ ( z ) {\displaystyle {\widehat {F}}(z)} for some fixed or formal z {... | wikipedia |
generating function transformation | Additionally, we use the bracket notation for coefficient extraction from the Concrete Mathematics reference which is given by F ( z ) := f n {\displaystyle F(z):=f_{n}} . The main article gives examples of generating functions for many sequences. Other examples of generating function variants include Dirichlet genera... | wikipedia |
unary function | In mathematics, a unary function is a function that takes one argument. A unary operator belongs to a subset of unary functions, in that its range coincides with its domain. In contrast, a unary function's domain may or may not coincide with its range. | wikipedia |
unary functional symbol | In mathematics, a unary operation is an operation with only one operand, i.e. a single input. This is in contrast to binary operations, which use two operands. An example is any function f: A → A, where A is a set. | wikipedia |
unary functional symbol | The function f is a unary operation on A. Common notations are prefix notation (e.g. ¬, −), postfix notation (e.g. factorial n! ), functional notation (e.g. sin x or sin(x)), and superscripts (e.g. transpose AT). Other notations exist as well, for example, in the case of the square root, a horizontal bar extending the ... | wikipedia |
properties of polynomial roots | In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots, if counted with their multiplicities. They form a multiset of n points in the complex plane. This article concerns the geometry of these points, that is the information about their localization in the complex plan... | wikipedia |
properties of polynomial roots | Such bounds are widely used for root-finding algorithms for polynomials, either for tuning them, or for computing their computational complexity. Some other properties are probabilistic, such as the expected number of real roots of a random polynomial of degree n with real coefficients, which is less than 1 + 2 π ln ... | wikipedia |
universal c*-algebra | In mathematics, a universal C*-algebra is a C*-algebra described in terms of generators and relations. In contrast to rings or algebras, where one can consider quotients by free rings to construct universal objects, C*-algebras must be realizable as algebras of bounded operators on a Hilbert space by the Gelfand-Naimar... | wikipedia |
wonders of numbers | In mathematics, a vampire number or true vampire number is a composite natural number v, with an even number of digits n, that can be factored into two integers x and y each with n/2 digits and not both with trailing zeroes, where v contains all the digits from x and from y, in any order. x and y are called the fangs. ... | wikipedia |
wonders of numbers | Similarly, 136,948 is a vampire because 136,948 = 146 × 938. Vampire numbers first appeared in a 1994 post by Clifford A. Pickover to the Usenet group sci.math, and the article he later wrote was published in chapter 30 of his book Keys to Infinity.In addition to "Vampire numbers", a term Pickover actually coined, he h... | wikipedia |
wonders of numbers | In 1990, he asked "Is There a Double Smoothly Undulating Integer? ", and he computed "All Known Replicating Fibonacci Digits Less than One Billion". With his colleague John R. Hendricks, he was the first to compute the smallest perfect (nasik) magic tesseract. The "Pickover sequence" dealing with e and pi was named aft... | wikipedia |
variable (logics) | In mathematics, a variable (from Latin variabilis, "changeable") is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a set.Algebraic computations with variables as if they were explicit numbers solve a ... | wikipedia |
vertex algebra | In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex operator algebras have proven useful in purely mathematical contexts such as monstrous moonshine and the geomet... | wikipedia |
vertex algebra | Borcherds formulated the notion of vertex algebra by axiomatizing the relations between the lattice vertex operators, producing an algebraic structure that allows one to construct new Lie algebras by following Frenkel's method. The notion of vertex operator algebra was introduced as a modification of the notion of vert... | wikipedia |
vertex algebra | Motivated by this observation, they added the Virasoro action and bounded-below property as axioms. We now have post-hoc motivation for these notions from physics, together with several interpretations of the axioms that were not initially known. | wikipedia |
vertex algebra | Physically, the vertex operators arising from holomorphic field insertions at points in two-dimensional conformal field theory admit operator product expansions when insertions collide, and these satisfy precisely the relations specified in the definition of vertex operator algebra. Indeed, the axioms of a vertex opera... | wikipedia |
von neumann algebras | In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra. Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of s... | wikipedia |
von neumann algebras | Two basic examples of von Neumann algebras are as follows: The ring L ∞ ( R ) {\displaystyle L^{\infty }(\mathbb {R} )} of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space L 2 (... | wikipedia |
von neumann algebras | Introductory accounts of von Neumann algebras are given in the online notes of Jones (2003) and Wassermann (1991) and the books by Dixmier (1981), Schwartz (1967), Blackadar (2005) and Sakai (1971). The three volume work by Takesaki (1979) gives an encyclopedic account of the theory. The book by Connes (1994) discusses... | wikipedia |
lie algebra bundle | In mathematics, a weak Lie algebra bundle ξ = ( ξ , p , X , θ ) {\displaystyle \xi =(\xi ,p,X,\theta )\,} is a vector bundle ξ {\displaystyle \xi \,} over a base space X together with a morphism θ: ξ ⊗ ξ → ξ {\displaystyle \theta :\xi \otimes \xi \rightarrow \xi } which induces a Lie algebra structure on each fibre ξ x... | wikipedia |
lie algebra bundle | Let denote the Lie bracket of s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} and deform it by the real parameter as: x = x ⋅ {\displaystyle _{x}=x\cdot } for X , Y ∈ s o ( 3 ) {\displaystyle X,Y\in {\mathfrak {so}}(3)} and x ∈ R {\displaystyle x\in \mathbb {R} } . Lie's third theorem states that every bundle of Lie a... | wikipedia |
lie algebra bundle | In general globally the total space might fail to be Hausdorff. But if all fibres of a real Lie algebra bundle over a topological space are mutually isomorphic as Lie algebras, then it is a locally trivial Lie algebra bundle. This result was proved by proving that the real orbit of a real point under an algebraic group... | wikipedia |
lie algebra bundle | Suppose the base space is Hausdorff and fibers of total space are isomorphic as Lie algebras then there exists a Hausdorff Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle. Every semi simple Lie algebra bundle is locally trivial. Hence there exist a Hausdo... | wikipedia |
web (differential geometry) | In mathematics, a web permits an intrinsic characterization in terms of Riemannian geometry of the additive separation of variables in the Hamilton–Jacobi equation. | wikipedia |
zero of a function | In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f {\displaystyle f} , is a member x {\displaystyle x} of the domain of f {\displaystyle f} such that f ( x ) {\displaystyle f(x)} vanishes at x {\displaystyle x} ; that is, the function f {\displaystyle f} at... | wikipedia |
zero of a function | For example, the polynomial f {\displaystyle f} of degree two, defined by f ( x ) = x 2 − 5 x + 6 {\displaystyle f(x)=x^{2}-5x+6} has the two roots (or zeros) that are 2 and 3. If the function maps real numbers to real numbers, then its zeros are the x {\displaystyle x} -coordinates of the points where its graph meets ... | wikipedia |
zeta function | In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}.} Zeta functions include: Airy zeta function, related to the zeros of the Airy function Arakawa–Kaneko zeta fu... | wikipedia |
zonal spherical function | In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficie... | wikipedia |
zonal spherical function | The spectrum of the commutative Banach *-algebra of biinvariant L1 functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series. Zonal spherical ... | wikipedia |
zonal spherical function | For complex groups, the theory simplifies significantly, because G is the complexification of K, and the formulas are related to analytic continuations of the Weyl character formula on K. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement. Apart from their group t... | wikipedia |
affine geometry | In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle. As the notion of parallel lines is one of the main properties that is independent of any metric, affine geometry is often considered as the study of paral... | wikipedia |
affine geometry | Comparisons of figures in affine geometry are made with affine transformations, which are mappings that preserve alignment of points and parallelism of lines. Affine geometry can be developed in two ways that are essentially equivalent.In synthetic geometry, an affine space is a set of points to which is associated a s... | wikipedia |
affine geometry | In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point t... | wikipedia |
algebraic cobordism | In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi-projective schemes over a field. It was introduced by Marc Levine and Fabien Morel (2001, 2001b). An oriented cohomology theory on the category of smooth quasi-projective schemes Sm over a field k consists of a contravariant functo... | wikipedia |
algebraic cobordism | In particular they are "oriented", which means roughly that they behave well on vector bundles; this is closely related to the condition that a generalized cohomology theory has a complex orientation. Over a field of characteristic 0, algebraic cobordism is the universal oriented cohomology theory for smooth varieties.... | wikipedia |
algebraic space | In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using th... | wikipedia |
pure injective module | In mathematics, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The solutions to these systems allow the extension of certain kinds of module homomorphisms. Thes... | wikipedia |
albert algebra | In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism. One of them, which was ... | wikipedia |
albert algebra | Another is defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution. Over any algebraically closed field, there is just one Albert algebra, and its automorphism group G is the simple split group of type F4. | wikipedia |
albert algebra | (For example, the complexifications of the three Albert algebras over the real numbers are isomorphic Albert algebras over the complex numbers.) Because of this, for a general field F, the Albert algebras are classified by the Galois cohomology group H1(F,G).The Kantor–Koecher–Tits construction applied to an Albert alg... | wikipedia |
artin l-function | In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below,... | wikipedia |
azumaya algebra | In mathematics, an Azumaya algebra is a generalization of central simple algebras to R-algebras where R need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where R is a commutative local ring. The notion was developed further in ring theory, and in algebraic geometry, where A... | wikipedia |
engel subalgebra | In mathematics, an Engel subalgebra of a Lie algebra with respect to some element x is the subalgebra of elements annihilated by some power of ad x. Engel subalgebras are named after Friedrich Engel. For finite-dimensional Lie algebras over infinite fields the minimal Engel subalgebras are the Cartan subalgebras. | wikipedia |
igusa zeta-function | In mathematics, an Igusa zeta function is a type of generating function, counting the number of solutions of an equation, modulo p, p2, p3, and so on. | wikipedia |
l function | In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An L-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation. The Riemann zeta function is an example... | wikipedia |
l function | The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of pro... | wikipedia |
n-topological space | In mathematics, an N-topological space is a set equipped with N arbitrary topologies. If τ1, τ2, ..., τN are N topologies defined on a nonempty set X, then the N-topological space is denoted by (X,τ1,τ2,...,τN). For N = 1, the structure is simply a topological space. For N = 2, the structure becomes a bitopological spa... | wikipedia |
o*-algebra | In mathematics, an O*-algebra is an algebra of possibly unbounded operators defined on a dense subspace of a Hilbert space. The original examples were described by Borchers (1962) and Uhlmann (1962), who studied some examples of O*-algebras, called Borchers algebras, arising from the Wightman axioms of quantum field th... | wikipedia |
ockham algebras | In mathematics, an Ockham algebra is a bounded distributive lattice with a dual endomorphism, that is, an operation ~ satisfying ~(x ∧ y) = ~x ∨ ~y, ~(x ∨ y) = ~x ∧ ~y, ~0 = 1, ~1 = 0. They were introduced by Berman (1977), and were named after William of Ockham by Urquhart (1979). Ockham algebras form a variety. Examp... | wikipedia |
oper (mathematics) | In mathematics, an Oper is a principal connection, or in more elementary terms a type of differential operator. They were first defined and used by Vladimir Drinfeld and Vladimir Sokolov to study how the KdV equation and related integrable PDEs correspond to algebraic structures known as Kac–Moody algebras. Their moder... | wikipedia |
rvachev function | In mathematics, an R-function, or Rvachev function, is a real-valued function whose sign does not change if none of the signs of its arguments change; that is, its sign is determined solely by the signs of its arguments.Interpreting positive values as true and negative values as false, an R-function is transformed into... | wikipedia |
homological algebra | In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative attempt to unify sever... | wikipedia |
homological algebra | The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well. These stability properties make them inevitable in homological algebra and... | wikipedia |
homological algebra | Abelian categories are named after Niels Henrik Abel. More concretely, a category is abelian if it has a zero object, it has all binary products and binary coproducts, and it has all kernels and cokernels. all monomorphisms and epimorphisms are normal. | wikipedia |
absolutely integrable | In mathematics, an absolutely integrable function is a function whose absolute value is integrable, meaning that the integral of the absolute value over the whole domain is finite. For a real-valued function, since where both ∫ f + ( x ) d x {\textstyle \int f^{+}(x)\,dx} and ∫ f − ( x ) d x {\textstyle \int f^{-}(x)\,... | wikipedia |
absolutely integrable | Let us define where ℜ f ( x ) {\displaystyle \Re f(x)} and ℑ f ( x ) {\displaystyle \Im f(x)} are the real and imaginary parts of f ( x ) {\displaystyle f(x)} . Then so This shows that the sum of the four integrals (in the middle) is finite if and only if the integral of the absolute value is finite, and the function i... | wikipedia |
adjunction space | In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y. Let f: A → X be a continuous map (called the attaching map). One forms the adjunct... | wikipedia |
admissible algebra | In mathematics, an admissible algebra is a (possibly non-associative) commutative algebra whose enveloping Lie algebra of derivations splits into the sum of an even and an odd part. Admissible algebras were introduced by Koecher (1967). | wikipedia |
affine hecke algebra | In mathematics, an affine Hecke algebra is the algebra associated to an affine Weyl group, and can be used to prove Macdonald's constant term conjecture for Macdonald polynomials. | wikipedia |
affine kac–moody algebra | In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, af... | wikipedia |
affine kac–moody algebra | As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simp... | wikipedia |
affine kac–moody algebra | More generally, if σ is an automorphism of the simple Lie algebra g {\displaystyle {\mathfrak {g}}} associated to an automorphism of its Dynkin diagram, the twisted loop algebra L σ g {\displaystyle L_{\sigma }{\mathfrak {g}}} consists of g {\displaystyle {\mathfrak {g}}} -valued functions f on the real line which sati... | wikipedia |
algebraic curves | In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenizin... | wikipedia |
algebraic curves | These two operations are each inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. More generally, an algebraic curve is an algebraic variety of dimension one. Equivalently, an algebraic curve is... | wikipedia |
algebraic curves | If the curve is contained in an affine space or a projective space, one can take a projection for such a birational equivalence. These birational equivalences reduce most of the study of algebraic curves to the study of algebraic plane curves. However, some properties are not kept under birational equivalence and must ... | wikipedia |
algebraic curves | This is, in particular, the case for the degree and smoothness. For example, there exist smooth curves of genus 0 and degree greater than two, but any plane projection of such curves has singular points (see Genus–degree formula). A non-plane curve is often called a space curve or a skew curve. | wikipedia |
algebra bundle | In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that the transition functions are algebra isomorphisms. Since algebras are also vector spaces, every algebra bundle is a vector bundle. Examples include the tensor-algebra bu... | wikipedia |
algebra homomorphism | In mathematics, an algebra homomorphism is a homomorphism between two algebras. More precisely, if A and B are algebras over a field (or a ring) K, it is a function F: A → B {\displaystyle F\colon A\to B} such that, for all k in K and x, y in A, one has F ( k x ) = k F ( x ) {\displaystyle F(kx)=kF(x)} F ( x + y ) = F ... | wikipedia |
algebra (ring theory) | In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms i... | wikipedia |
algebra (ring theory) | An algebra is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order n forms a unital algebra since the identity matrix of order n is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a (un... | wikipedia |
algebra (ring theory) | Many authors use the term algebra to mean associative algebra, or unital associative algebra, or in some subjects such as algebraic geometry, unital associative commutative algebra. Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. Algebras are not to be co... | wikipedia |
homotopy associative algebra | In mathematics, an algebra such as ( R , + , ⋅ ) {\displaystyle (\mathbb {R} ,+,\cdot )} has multiplication ⋅ {\displaystyle \cdot } whose associativity is well-defined on the nose. This means for any real numbers a , b , c ∈ R {\displaystyle a,b,c\in \mathbb {R} } we have a ⋅ ( b ⋅ c ) − ( a ⋅ b ) ⋅ c = 0 {\displaysty... | wikipedia |
homotopy associative algebra | The study of A ∞ {\displaystyle A_{\infty }} -algebras is a subset of homotopical algebra, where there is a homotopical notion of associative algebras through a differential graded algebra with a multiplication operation and a series of higher homotopies giving the failure for the multiplication to be associative. Loos... | wikipedia |
homotopy associative algebra | When looking at the underlying cohomology algebra H ( A ∙ , m 1 ) {\displaystyle H(A^{\bullet },m_{1})} , the map m 2 {\displaystyle m_{2}} should be an associative map. Then, these higher maps m 3 , m 4 , … {\displaystyle m_{3},m_{4},\ldots } should be interpreted as higher homotopies, where m 3 {\displaystyle m_{3}} ... | wikipedia |
homotopy associative algebra | Their structure was originally discovered by Jim Stasheff while studying A∞-spaces, but this was interpreted as a purely algebraic structure later on. These are spaces equipped with maps that are associative only up to homotopy, and the A∞ structure keeps track of these homotopies, homotopies of homotopies, and so fort... | wikipedia |
algebraic cycle | In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the varie... | wikipedia |
algebraic cycle | The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann... | wikipedia |
algebraic cycle | While divisors on higher-dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider. The behavior of these cycles is strikingly different from that of divisors. For example, every curve ... | wikipedia |
algebraic cycle | The hypothesis that the geometric genus is positive essentially means (by the Lefschetz theorem on (1,1)-classes) that the cohomology group H 2 ( S ) {\displaystyle H^{2}(S)} contains transcendental information, and in effect Mumford's theorem implies that, despite CH 2 ( S ) {\displaystyle \operatorname {CH} ^{2}(S)... | wikipedia |
algebraic cycle | The Hodge conjecture, one of the Clay Mathematics Institute's Millennium Prize Problems, predicts that the topology of a complex algebraic variety forces the existence of certain algebraic cycles. The Tate conjecture makes a similar prediction for étale cohomology. Alexander Grothendieck's standard conjectures on algeb... | wikipedia |
polynomial equation | In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0 {\displaystyle P=0} where P is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term algebraic equation refers only to univariate equations, that is polynomial equati... | wikipedia |
polynomial equation | For example, x 5 − 3 x + 1 = 0 {\displaystyle x^{5}-3x+1=0} is an algebraic equation with integer coefficients and y 4 + x y 2 − x 3 3 + x y 2 + y 2 + 1 7 = 0 {\displaystyle y^{4}+{\frac {xy}{2}}-{\frac {x^{3}}{3}}+xy^{2}+y^{2}+{\frac {1}{7}}=0} is a multivariate polynomial equation over the rationals. Some but not all... | wikipedia |
algebraic expression | In mathematics, an algebraic expression is an expression built up from constant algebraic numbers, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, 3x2 − 2xy + c is an algebraic expression. Since taking th... | wikipedia |
algebraic expression | Usually, π is constructed as a geometric relationship, and the definition of e requires an infinite number of algebraic operations. A rational expression is an expression that may be rewritten to a rational fraction by using the properties of the arithmetic operations (commutative properties and associative properties ... | wikipedia |
algebraic expression | Thus, 3 x − 2 x y + c y − 1 {\displaystyle {\frac {3x-2xy+c}{y-1}}} is a rational expression, whereas 1 − x 2 1 + x 2 {\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}} is not. A rational equation is an equation in which two rational fractions (or rational expressions) of the form P ( x ) Q ( x ) {\displaystyle {\frac ... | wikipedia |
algebraic expression | These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected. | wikipedia |
algebraic extension | In mathematics, an algebraic extension is a field extension L/K such that every element of the larger field L is algebraic over the smaller field K; that is, every element of L is a root of a non-zero polynomial with coefficients in K. A field extension that is not algebraic, is said to be transcendental, and must cont... | wikipedia |
algebraic function field | In mathematics, an algebraic function field (often abbreviated as function field) of n variables over a field k is a finitely generated field extension K/k which has transcendence degree n over k. Equivalently, an algebraic function field of n variables over k may be defined as a finite field extension of the field K =... | wikipedia |
algebraic function | In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addition, subtraction, multiplication, division, and raising to a fractional pow... | wikipedia |
algebraic function | It can be shown that the same class of functions is obtained if algebraic numbers are accepted for the coefficients of the ai(x)'s. If transcendental numbers occur in the coefficients the function is, in general, not algebraic, but it is algebraic over the field generated by these coefficients. The value of an algebrai... | wikipedia |
algebraic function | Sometimes, coefficients a i ( x ) {\displaystyle a_{i}(x)} that are polynomial over a ring R are considered, and one then talks about "functions algebraic over R". A function which is not algebraic is called a transcendental function, as it is for example the case of exp x , tan x , ln x , Γ ( x ) {\displaystyle ... | wikipedia |
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